On The Integer Solutions of The Diophantine Equations x^2-(a^2 b^2+b) y^2-(4c+2)x+4(a^2 b^2+b)y-4(a^2 b^2+b-c^2-c)=0

Abstract

Using the theory of Pellian equations, we show that the Diophantine equations    have infinitely many nontrivial integer solutions (x, y) by using Generalized Biperiodic Fibonacci and lucas sequence. We also derive some recurrence relations on the integer solutions (x, y) of E. AMS Subject Classification: 11D09.